Theorem crnggrpd 20222

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20221	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20217	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18903  CRingccrg 20209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6449  df-fv 6501  df-ov 7364  df-ring 20210  df-cring 20211

This theorem is referenced by:  fermltlchr  21522  psdmul  22145  psd1  22146  psdpw  22149  ply1chr  22284  ply1fermltlchr  22290  elrgspnsubrunlem1  33326  elrgspnsubrunlem2  33327  erlbr2d  33343  rlocaddval  33347  rloccring  33349  rloc0g  33350  rlocf1  33352  fracerl  33385  gsumind  33423  ressply1evls1  33643  evl1deg1  33654  evl1deg2  33655  evl1deg3  33656  ply1dg1rt  33658  vr1nz  33671  psrmonprod  33714  mplmonprod  33716  esplyfvn  33739  irngss  33850  extdgfialglem1  33855  irredminply  33879  algextdeglem4  33883  algextdeglem5  33884  aks6d1c1p3  42566  aks6d1c2lem4  42583  aks6d1c6lem2  42627  aks5lem2  42643  evlsbagval  43019  selvvvval  43035  evlselv  43037  selvadd  43038  prjcrv0  43083